17.0 Linear Regression


 Lesley Morris Cox
 1 years ago
 Views:
Transcription
1 17.0 Linear Regression 1 Answer Questions Lines Correlation Regression
2 17.1 Lines The algebraic equation for a line is Y = β 0 + β 1 X 2 The use of coordinate axes to show functional relationships was invented by René Descartes ( ). He was an artillery officer, and probably got the idea from pictures that showed the trajectories of cannonballs.
3 17.2 Correlation 3 Sir Francis Galton explored Africa, invented eugenics, studied whether ships that carried missionaries were less likely to be lost at sea, pioneered birthanddeath models and meteorology, and was Charles Darwin s cousin. He also was the first to conceive of linear regression (although he did not have the mathematical skill to develop the formulae, and got a friend of his at Cambridge to do the derivations).
4 Correlation is a measure of the strength of the linear association between two continuous variables. An early example studied the relationship between the height of fathers and the height of sons. 4 Clearly, tall fathers tend to have tall sons, and short fathers tend to have short sons. If the father s height were a perfect predictor of the son s height, then all fatherson pairs would lie on a straight line in a scatterplot. Regression fits a line to the points in a scatterplot. The term comes from the fatherson example. An exceptionally tall father tends to have sons that are shorter than himself; an exceptionally short father tends to have sons that are taller than himself. Thus the sons height tend to regress towards the mean.
5 The sample correlation coefficient r measures the strength of the linear association between X and Y values in a scatterplot. If the absolute value of the correlation is near 1, then knowing one variable determines the other variable almost perfectly (if the relationship is linear). r lies between 1 and 1, inclusive. 5 r equals 1 iff all points lie on a line with positive slope. r equals 1 iff all points lie on a line with negative slope. nonzero r does not imply a causal relationship. The square of the correlation is called the coefficient of determination. It is the proportion of the variation in Y that is explained by knowledge of X.
6 6
7 To estimate the true correlation coefficient, define SS xx = (x i x) 2 = x 2 i n x 2 SS yy = (y i ȳ) 2 = y 2 i nȳ 2 SS xy = (x i x)(y i ȳ) = x i y i n xȳ. 7 Note: if divided by n 1, these are the sample versions of the variances and the covariance. So there s no need to memorize. Then the sample correlation is r = SS xy SSxx SS yy. One can show that the coefficient of determination r 2 is the proportion of the variance in Y that is explained by knowledge of X.
8 Correlations are often high when some factor affects both X and Y. GPA and SAT scores are both affected by IQ. number of hours spent listening to Rob Zombie and GPA are both affected by lifestyle. 8 It is hard to argue that correlation implies causation. GPA does not cause SAT, and Rob Zombie does not hurt GPA. But sometimes, there might be a causal link. Hours of study are probably correlated with GPA, and it seems likely to be causal. Ecological correlations occur when X or Y or both is an average, proportion, or a percentage for a group. Here causation is especially difficult to show. The original link between smoking and lung cancer was an ecological correlation (Doll, 1955). The scatterplot showed the lung cancer rate against the proportion of smokers for 11 different countries.
9 9
10 17.3 Regression The mathematical model for regression assumes that: 1. Each point (X i, Y i ) in the scatterplot satisfies: Y i = β 0 + β 1 X i + ǫ i 10 where the ǫ i have a normal distribution with mean zero and (usually) unknown standard deviation. 2. The errors ǫ i are independent. 3. The X i values are measured without error. (Thus all error occurs in the vertical direction.) The response variable is labeled Y. This is sometimes called the dependent variable. The explanatory variable is labeled X. This is sometimes called the independent variable, or the covariate.
11 Regression tries to fit the best straight line to the data. Specifically, it fits the line that minimizes the sum of the squared deviations from each point to the line, where deviation is measured in the vertical direction. Note: This does not measure deviation as the perpendicular distance from the point to the line. 11
12 How does one find the estimates ˆβ 0 and ˆβ 1 of the coefficients in the regression equation? We need to get the values that minimize the sum of the squared vertical distances. (Gauss, of course.) The sum of the squared vertical distances is 12 f(β 0, β 1 ) = n [Y i (β 0 + β 1 X i )] 2. i=1 So take the derivative of f(β 0, β 1 ) with respect to β 0 and β 1, set these equal to zero, and solve. One finds that: ˆβ 0 = Ȳ ˆβ 1 X; ˆβ1 = SS xy /SS xx.
13 The regression line predicts the average value of Y for a specific value of X. This is not the same as saying that an individual s value lies on the line. An individual is likely to be far from the line. 13 Under our assumptions, the distance of an individual from the regression line is normally distributed with mean 0 and standard deviation σ ǫ. We do not know the true σ ǫ, but we can estimate it from the sample standard deviation of the residuals. The residuals are the {ˆǫ i = y i ŷ i }, where ŷ i is the value predicted by the regression line. This difference is the estimated error ˆǫ i for the ith observation. Then ˆσ ǫ = 1 n (y i ŷ i ) n 2 2. Why do we divide by n 2? i=1
14 Recall that SS x = n i=1 (X i X) 2. Then a twosided 100(1 α)% confidence interval on the location of the true regression line at x is ˆβ 0 + ˆβ 1 (x X) 2 1 x ± ˆσ ǫ + t n 2,α/2. n SS x 14 A twosided 100(1 α)% prediction interval on the location of an individual whose value of the explanatory variable is x is ˆβ 0 + ˆβ 1 x ± ˆσ ǫ (x X) 2 + t n 2,α/2. n SS x Onesided intervals are formed in the obvious way. If the sample size is large, you use the ztable instead of the t n 2 table. And if you sample without replacement, you can use the FPCF to multiply ˆσ ǫ. The ˆσ ǫ is sometimes called root mean squared error or rmse.
15 Example 1.a: The DUS want to set 95% confidence intervals on the starting salaries (in thousands) of Duke statistics majors as a function of their GPA. Based on the 15 people who majored last year, we find β 0 = 20 and ˆβ 1 = 10. The rmse was 4, SS x = 4, X = 3.2. What is the average starting salary for people who have GPAs of 3.5? 15 ˆβ 0 + ˆβ 1 1 x ± ˆσ ǫ n (x X) 2 + SS x t n 2,α/2 = ± ( ) The DUS is 95% confident that the mean starting salary is between L = $52.42K and U = $57.58K.
16 Example 1.b: Poindexter has a GPA of 3.5 and asks the DUS to set a 95% prediction interval on his starting salary. 16 ˆβ 0 + ˆβ 1 x ± ˆσ ǫ n = ± 4 (x X) 2 + SS x t n 2,α/2 + ( ) The DUS is 95% confident that his starting salary will be between L = $45.98K and U = $64.02K. Note that for both intervals, the uncertainty increases as one tries to set intervals for xvalues that are far from X. This is reasonable if there is a certain amount of wigggle error in the fitted regression line, the magnitude of the error increases with distance from X.
17 Be aware that regressing weight as a function of height gives a different regression line than regressing height against weight. If your best estimate of the weight of a man who is 5 10 is 170 pounds, that does not mean that the best estimate of the height of a man who weighs 170 pounds is The regression fallacy mistakenly argues that there is some effect or force that causes sons to be more average than their fathers. In fact, this is only the natural operation of random chance. Consider scores on a first and second exam, and also the fatherson height example. What can you say about the performance of baseball players in the first and second halves of the season? Or stocktraders, or new employees?
e = random error, assumed to be normally distributed with mean 0 and standard deviation σ
1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.
More informationChapter 7: Simple linear regression Learning Objectives
Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) 
More informationHypothesis testing  Steps
Hypothesis testing  Steps Steps to do a twotailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationLecture 18 Linear Regression
Lecture 18 Statistics Unit Andrew Nunekpeku / Charles Jackson Fall 2011 Outline 1 1 Situation  used to model quantitative dependent variable using linear function of quantitative predictor(s). Situation
More informationThe aspect of the data that we want to describe/measure is the degree of linear relationship between and The statistic r describes/measures the degree
PS 511: Advanced Statistics for Psychological and Behavioral Research 1 Both examine linear (straight line) relationships Correlation works with a pair of scores One score on each of two variables ( and
More information2. Simple Linear Regression
Research methods  II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More informationwhere b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.
Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes
More informationRegression. In this class we will:
AMS 5 REGRESSION Regression The idea behind the calculation of the coefficient of correlation is that the scatter plot of the data corresponds to a cloud that follows a straight line. This idea can be
More informationRegression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture  2 Simple Linear Regression
Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture  2 Simple Linear Regression Hi, this is my second lecture in module one and on simple
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationAMS7: WEEK 8. CLASS 1. Correlation Monday May 18th, 2015
AMS7: WEEK 8. CLASS 1 Correlation Monday May 18th, 2015 Type of Data and objectives of the analysis Paired sample data (Bivariate data) Determine whether there is an association between two variables This
More informationUnivariate Regression
Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is
More informationExample: Boats and Manatees
Figure 96 Example: Boats and Manatees Slide 1 Given the sample data in Table 91, find the value of the linear correlation coefficient r, then refer to Table A6 to determine whether there is a significant
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More information, has mean A) 0.3. B) the smaller of 0.8 and 0.5. C) 0.15. D) which cannot be determined without knowing the sample results.
BA 275 Review Problems  Week 9 (11/20/0611/24/06) CD Lessons: 69, 70, 1620 Textbook: pp. 520528, 111124, 133141 An SRS of size 100 is taken from a population having proportion 0.8 of successes. An
More informationRegression Analysis: Basic Concepts
The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
More informationCoefficient of Determination
Coefficient of Determination The coefficient of determination R 2 (or sometimes r 2 ) is another measure of how well the least squares equation ŷ = b 0 + b 1 x performs as a predictor of y. R 2 is computed
More informationAnswer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade
Statistics Quiz Correlation and Regression  ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements
More information12/31/2016. PSY 512: Advanced Statistics for Psychological and Behavioral Research 2
PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand linear regression with a single predictor Understand how we assess the fit of a regression model Total Sum of Squares
More informationStatistiek II. John Nerbonne. March 24, 2010. Information Science, Groningen Slides improved a lot by Harmut Fitz, Groningen!
Information Science, Groningen j.nerbonne@rug.nl Slides improved a lot by Harmut Fitz, Groningen! March 24, 2010 Correlation and regression We often wish to compare two different variables Examples: compare
More informationLecture 5: Correlation and Linear Regression
Lecture 5: Correlation and Linear Regression 3.5. (Pearson) correlation coefficient The correlation coefficient measures the strength of the linear relationship between two variables. The correlation is
More information, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (
Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we
More informationSimple linear regression
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
More informationCorrelation. What Is Correlation? Perfect Correlation. Perfect Correlation. Greg C Elvers
Correlation Greg C Elvers What Is Correlation? Correlation is a descriptive statistic that tells you if two variables are related to each other E.g. Is your related to how much you study? When two variables
More informationInference for Regression
Simple Linear Regression Inference for Regression The simple linear regression model Estimating regression parameters; Confidence intervals and significance tests for regression parameters Inference about
More information17. SIMPLE LINEAR REGRESSION II
17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.
More informationLesson Lesson Outline Outline
Lesson 15 Linear Regression Lesson 15 Outline Review correlation analysis Dependent and Independent variables Least Squares Regression line Calculating l the slope Calculating the Intercept Residuals and
More informationSession 7 Bivariate Data and Analysis
Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table covariation least squares
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More information. 58 58 60 62 64 66 68 70 72 74 76 78 Father s height (inches)
PEARSON S FATHERSON DATA The following scatter diagram shows the heights of 1,0 fathers and their fullgrown sons, in England, circa 1900 There is one dot for each fatherson pair Heights of fathers and
More informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationSTT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012)
STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012) TA: Zhen (Alan) Zhang zhangz19@stt.msu.edu Office hour: (C500 WH) 1:45 2:45PM Tuesday (office tel.: 4323342) Helproom: (A102 WH) 11:20AM12:30PM,
More informationThe correlation coefficient
The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative
More informationSection 14 Simple Linear Regression: Introduction to Least Squares Regression
Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship
More informationModule 3: Correlation and Covariance
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
More informationLinear Regression. Chapter 5. Prediction via Regression Line Number of new birds and Percent returning. Least Squares
Linear Regression Chapter 5 Regression Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y). We can then predict the average response for all subjects
More informationRegression III: Advanced Methods
Lecture 5: Linear leastsquares Regression III: Advanced Methods William G. Jacoby Department of Political Science Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Simple Linear Regression
More information12/31/2016. PSY 512: Advanced Statistics for Psychological and Behavioral Research 2
PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand when to use multiple Understand the multiple equation and what the coefficients represent Understand different methods
More information5. Linear Regression
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
More informationChapter 10  Practice Problems 1
Chapter 10  Practice Problems 1 1. A researcher is interested in determining if one could predict the score on a statistics exam from the amount of time spent studying for the exam. In this study, the
More informationCorrelation key concepts:
CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)
More informationAP Statistics 2002 Scoring Guidelines
AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought
More informationInstructor s Manual Prepared by Holly Raffle
Instructor s Manual Prepared by Holly Raffle Table of Contents Section 1 Introduction and Features 2 Section 2 Installation and Setup 3 Section 3 Using the Instructor s Manual 4 Section 4 Analyzing One
More informationHomework 11. Part 1. Name: Score: / null
Name: Score: / Homework 11 Part 1 null 1 For which of the following correlations would the data points be clustered most closely around a straight line? A. r = 0.50 B. r = 0.80 C. r = 0.10 D. There is
More informationRegression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between
More information1 Simple Linear Regression I Least Squares Estimation
Simple Linear Regression I Least Squares Estimation Textbook Sections: 8. 8.3 Previously, we have worked with a random variable x that comes from a population that is normally distributed with mean µ and
More informationCorrelation & Regression, II. Residual Plots. What we like to see: no pattern. Steps in regression analysis (so far)
Steps in regression analysis (so far) Correlation & Regression, II 9.07 4/6/2004 Plot a scatter plot Find the parameters of the best fit regression line, y =a+bx Plot the regression line on the scatter
More informationLecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation
Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Display and Summarize Correlation for Direction and Strength Properties of Correlation Regression Line Cengage
More information3.4 Statistical inference for 2 populations based on two samples
3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted
More informationChapter 9. Section Correlation
Chapter 9 Section 9.1  Correlation Objectives: Introduce linear correlation, independent and dependent variables, and the types of correlation Find a correlation coefficient Test a population correlation
More informationElementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination
Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used
More informationStatistics II Final Exam  January Use the University stationery to give your answers to the following questions.
Statistics II Final Exam  January 2012 Use the University stationery to give your answers to the following questions. Do not forget to write down your name and class group in each page. Indicate clearly
More informationChapter 13 Introduction to Linear Regression and Correlation Analysis
Chapter 3 Student Lecture Notes 3 Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
More informationAlgebra I: Lesson 54 (5074) SAS Curriculum Pathways
TwoVariable Quantitative Data: Lesson Summary with Examples Bivariate data involves two quantitative variables and deals with relationships between those variables. By plotting bivariate data as ordered
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informationExercise 1.12 (Pg. 2223)
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.
More informationSTAT 350 Practice Final Exam Solution (Spring 2015)
PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
More informationDESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.
DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,
More informationChapter 11: Two Variable Regression Analysis
Department of Mathematics Izmir University of Economics Week 1415 20142015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions
More informationRegression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology
Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of
More informationIntroduction to Linear Regression
14. Regression A. Introduction to Simple Linear Regression B. Partitioning Sums of Squares C. Standard Error of the Estimate D. Inferential Statistics for b and r E. Influential Observations F. Regression
More informationDescriptive statistics; Correlation and regression
Descriptive statistics; and regression Patrick Breheny September 16 Patrick Breheny STA 580: Biostatistics I 1/59 Tables and figures Descriptive statistics Histograms Numerical summaries Percentiles Human
More informationSection 3 Part 1. Relationships between two numerical variables
Section 3 Part 1 Relationships between two numerical variables 1 Relationship between two variables The summary statistics covered in the previous lessons are appropriate for describing a single variable.
More informationSimple Regression Theory I 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY I 1 Simple Regression Theory I 2010 Samuel L. Baker Regression analysis lets you use data to explain and predict. A simple regression line drawn through data points In Assignment
More information1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ
STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material
More informationStep 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:
In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
More informationStudy Resources For Algebra I. Unit 1C Analyzing Data Sets for Two Quantitative Variables
Study Resources For Algebra I Unit 1C Analyzing Data Sets for Two Quantitative Variables This unit explores linear functions as they apply to data analysis of scatter plots. Information compiled and written
More informationEXPERIMENT 6: HERITABILITY AND REGRESSION
BIO 184 Laboratory Manual Page 74 EXPERIMENT 6: HERITABILITY AND REGRESSION DAY ONE: INTRODUCTION TO HERITABILITY AND REGRESSION OBJECTIVES: Today you will be learning about some of the basic ideas and
More informationCorrelational Research. Correlational Research. Stephen E. Brock, Ph.D., NCSP EDS 250. Descriptive Research 1. Correlational Research: Scatter Plots
Correlational Research Stephen E. Brock, Ph.D., NCSP California State University, Sacramento 1 Correlational Research A quantitative methodology used to determine whether, and to what degree, a relationship
More informationIn Chapter 2, we used linear regression to describe linear relationships. The setting for this is a
Math 143 Inference on Regression 1 Review of Linear Regression In Chapter 2, we used linear regression to describe linear relationships. The setting for this is a bivariate data set (i.e., a list of cases/subjects
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x  x) B. x 3 x C. 3x  x D. x  3x 2) Write the following as an algebraic expression
More informationE205 Final: Version B
Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationStatistics 151 Practice Midterm 1 Mike Kowalski
Statistics 151 Practice Midterm 1 Mike Kowalski Statistics 151 Practice Midterm 1 Multiple Choice (50 minutes) Instructions: 1. This is a closed book exam. 2. You may use the STAT 151 formula sheets and
More informationChapter 12 : Linear Correlation and Linear Regression
Number of Faculty Chapter 12 : Linear Correlation and Linear Regression Determining whether a linear relationship exists between two quantitative variables, and modeling the relationship with a line, if
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 15 scale to 0100 scores When you look at your report, you will notice that the scores are reported on a 0100 scale, even though respondents
More informationDEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests
DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also
More informationClass 6: Chapter 12. Key Ideas. Explanatory Design. Correlational Designs
Class 6: Chapter 12 Correlational Designs l 1 Key Ideas Explanatory and predictor designs Characteristics of correlational research Scatterplots and calculating associations Steps in conducting a correlational
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationIntroduction to Regression. Dr. Tom Pierce Radford University
Introduction to Regression Dr. Tom Pierce Radford University In the chapter on correlational techniques we focused on the Pearson R as a tool for learning about the relationship between two variables.
More informationCOMP6053 lecture: Relationship between two variables: correlation, covariance and rsquared. jn2@ecs.soton.ac.uk
COMP6053 lecture: Relationship between two variables: correlation, covariance and rsquared jn2@ecs.soton.ac.uk Relationships between variables So far we have looked at ways of characterizing the distribution
More informationChapter 11: Linear Regression  Inference in Regression Analysis  Part 2
Chapter 11: Linear Regression  Inference in Regression Analysis  Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.
More informationCURVE FITTING LEAST SQUARES APPROXIMATION
CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship
More informationLesson 4 Part 1. Relationships between. two numerical variables. Correlation Coefficient. Relationship between two
Lesson Part Relationships between two numerical variables Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear between two numerical variables Relationship
More information0.1 Multiple Regression Models
0.1 Multiple Regression Models We will introduce the multiple Regression model as a mean of relating one numerical response variable y to two or more independent (or predictor variables. We will see different
More informationMCSE004. Dec 2013 Solutions Manual IGNOUUSER
MCSE004 Dec 2013 Solutions Manual IGNOUUSER 1 1. (a) Verify the distributive property of floating point numbers i.e. prove : a(bc) ab ac a=.5555e1, b=.4545e1, c=.4535e1 Define : Truncation error, Absolute
More informationUnit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationRegression Line. Guessing the Regression Line
1 Review Question (10 points) (a) What is the correlation coefficient for the data set below? x y 1 1 4 8 6 10 6 10 6 14 7 17 (b) If possible, fill in the blanks below so that the correlation will be equal
More informationCorrelation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables 2
Lesson 4 Part 1 Relationships between two numerical variables 1 Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationMind on Statistics. Chapter 3
Mind on Statistics Chapter 3 Section 3.1 1. Which one of the following is not appropriate for studying the relationship between two quantitative variables? A. Scatterplot B. Bar chart C. Correlation D.
More information